On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Abstract

A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any ntimes n system of conservation laws has a solution. The solution concept is an extension of the notion of singular delta -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of delta -distributions as a part of solutions to systems of conservation laws.